Does the constant 1 drop out in this very simple algebraic simplification

Question

I was integrating $\int \frac{x^3}{x^2+1}\, dx$, and my answer was

\begin{align} &\frac{x^2+1}{2}- \frac{\ln(x^2+1)}{2} + C\\& \text{ Rewrite/Simplify:}\\&=\frac{x^2-\ln(x^2+1)}{2} + C \end{align}

How exactly does the top simplify into the bottom? Shouldn't it be $x^2+1-ln(x^2+1)$? Why is there no $+1$?

Answer 1

Let $D= C+\frac12$ which is a constant.

\begin{align} \frac{x^2+1}{2}- \frac{\ln(x^2+1)}{2} + C=\frac{x^2-\ln(x^2+1)}{2} + D \end{align}

Here, they just reuse the notation and write $D$ as $C$. It is just an arbitrary constant.

Answer 2

Absorb $\frac12$ into the constant...